Int
ern
at
i
onal
Journ
al of Ele
ctrical
an
d
Co
mput
er
En
gi
n
eeri
ng
(IJ
E
C
E)
Vo
l.
10
,
No.
4
,
A
ugus
t
2020,
pp. 369
5~37
01
IS
S
N: 20
88
-
8708
,
DOI: 10
.11
591/
ijece
.
v10
i
4
.
pp3695
-
37
01
3695
Journ
al h
om
e
page
:
http:
//
ij
ece.i
aesc
or
e.c
om/i
nd
ex
.ph
p/IJ
ECE
Identity
-
based th
reshold
group si
gnatu
re
schem
e
based on
mu
ltip
le
hard num
ber theor
etic probl
ems
Ned
al Ta
hat
1
, Ashr
af A. Ta
h
at
2
1
Depa
rtment of
Mathe
m
at
i
cs,
Th
e
Hashem
it
e
Uni
ver
sit
y
,
Jordan
2
Depa
rtment of
Com
m
unic
at
ions E
ngineeri
ng
,
P
rinc
ess Sum
a
y
a Unive
rsit
y
for
T
ec
hnolog
y
,
Jord
an
Art
ic
le
In
f
o
ABSTR
A
CT
Art
ic
le
history:
Re
cei
ved
Oct
28
, 201
9
Re
vised
Jan
1
0
,
20
20
Accepte
d
Ja
n
29
, 2
020
W
e
int
roduc
e
in
thi
s
pape
r
a
new
ide
nti
t
y
-
b
ase
d
t
hre
shold
signat
u
re
(IBTHS)
te
chn
ique
,
whic
h
is
base
d
on
a
pai
r
of
in
tr
acta
b
le
prob
le
m
s,
resi
duosit
y
an
d
discre
t
e
logarith
m
.
Thi
s
techniqu
e
relies
on
two
d
iffi
cu
lt
probl
ems
and
offe
rs
an
improved
le
v
el
of
sec
urity
r
elati
v
e
to
an
on
t
wo
diffi
cul
t
har
d
proble
m
s
.
The
m
aj
ority
o
f
the
denot
ed
IBTHS
te
chni
q
ues
are
esta
b
li
shed
on
an
indi
vidual
diffic
ult
proble
m
.
De
spite
the
fa
ct
th
at
the
se
m
et
hod
s
are
sec
ure
,
howeve
r,
a
pros
pec
t
ive
solu
ti
on
of
th
is
sole
pr
oble
m
b
y
an
ad
ver
sar
y
wil
l
ena
bl
e
him/her
to
rec
over
the
e
nti
re
private
data
toge
ther
with
sec
ret
k
e
y
s
and
conf
igura
t
io
n
val
u
es
of
the
a
ss
oci
at
ed
sche
m
e.
Our
technique
is
imm
une
to
th
e
four
m
ost
familiar
at
t
ac
k
t
y
p
es
in
re
la
t
ion
to
the
signat
ur
e
sche
m
es.
Enha
nc
ed
per
fo
rm
anc
e
of
our
proposed
te
chnique
is
ver
ifi
ed
in
te
rm
s
of
m
ini
m
um
cost
o
f
computat
ions
r
equi
red
b
y
bo
th
of
the
sign
ing
algorithm
and
the
v
eri
f
y
ing
a
lg
orit
hm
in addi
t
io
n
to im
m
unity
to
attac
ks
.
Ke
yw
or
d
s
:
Discrete l
oga
rithm
Id
e
ntit
y
-
base
d si
gn
at
ur
e
Re
siduosit
y
Thr
e
shold
sig
na
ture
Copyright
©
202
0
Instit
ut
e
o
f Ad
vanc
ed
Engi
n
ee
r
ing
and
S
cienc
e
.
Al
l
rights re
serv
ed
.
Corres
pond
in
g
Aut
h
or
:
Ned
al
Tahat,
Dep
a
rtm
ent o
f M
at
hem
a
ti
cs,
The Has
hem
it
e
Unive
rsity
,
Zarq
a
13
133, J
orda
n.
Em
a
il
: ned
al
@
hu.edu.
jo
1.
INTROD
U
CTION
In
1971,
the
i
dea
of
dig
it
al
sign
at
ur
e
was
first
prese
nted
by
Di
ff
ie
a
nd
Helm
an
[1
]
t
hat
ena
ble
d
a
s
ign
er
in
po
s
session
of
a
se
crete
key
to
sign
a
m
essage,
wh
il
e
anyb
ody
us
ing
a
pu
blic
key
cou
ld
perform
ver
ific
at
io
n
of
the
sign
at
ure.
The
noti
on
of
thres
hold
sig
natu
res
was
presente
d
by
D
esm
edt
[2
]
in
1987.
A
secret
key,
and
c
orres
pondin
gly,
the
sig
ning
p
ow
e
r,
is
sh
are
d
to
a
colle
ct
ion
of
play
ers
in
a
(
,
)
thres
ho
l
d
sig
na
ture
sc
hem
e,
wh
e
re
this
is
a
ccom
plished
in
a
m
ann
er
t
ha
t
any
subset
of
play
ers
is
able
to
colle
ct
ively
deliver
a
sig
natu
r
e
on
t
he
acco
unt
of
t
he
gr
oup,
w
her
eas
a
s
ubset
com
po
se
d
o
f
up
t
o
−
1
pl
ay
ers
is
incapab
le
.
T
he
thres
hold
sign
at
ur
e
is
fun
dam
ental
yet
of
a
gr
eat
sign
ific
ance
crypt
ogra
ph
ic
sch
em
e
that
is
du
e
to
it
s
bif
ol
d
f
un
ct
io
n:
by
boos
ti
ng
the
oppo
rtu
nity
of
the
sig
ning
age
ncy
wh
il
e
sim
ultaneo
us
ly
i
m
pro
ving
the
safe
guar
di
ng
proce
ss
a
ga
inst
fr
a
udule
nc
e
thr
ough
c
omplet
ing
t
he
le
ar
n
process
of
th
e
secret
sig
nat
ur
e
key
for
the
anta
gonist.
S
ubseq
ue
nt
to
Desm
edt’s
creati
on,
i
n
the
com
m
on
ly
-
nam
ed
thr
esh
old
c
rypto
gr
a
phy
do
m
ai
n,
se
veral
thres
ho
l
d
si
gn
at
ur
e
ap
pro
aches
i
ncorpo
r
at
ed
on
di
ver
s
e
prem
ise
wer
e
f
or
m
ulate
d,
su
c
h
as
[3
-
8].
In
ord
er
to
stream
li
n
e
key
m
anag
e
m
ent
pr
oc
esses
in
certi
ficat
e
ba
sed
pu
blic
ke
y
set
ti
ng
,
Sh
a
m
ir
[9
]
in
19
84,
cal
le
d
for
i
den
ti
ty
-
ba
sed
(ID
-
base
d)
enc
ryptio
n
a
nd
sig
nat
ur
e
m
et
hods
.
The
nce
f
or
t
h,
i
n
the
sco
pe
o
f
this
com
m
on
ly
-
nam
ed
ID
-
ba
s
ed
crypt
ogra
phy,
sco
res
of
i
den
ti
ty
-
base
d
crypto
gr
a
phy
te
chn
i
qu
e
s
we
r
e
pu
t
forw
a
r
d,
s
uch
as
the
w
orks
of
[
10,
11
]
.
T
he
rem
ark
ably
pr
om
inent
too
l
ha
s
pro
posed
bi
li
near
pai
rin
g
[
10
]
in
const
ru
ct
in
g
identit
y
-
base
d
crypto
gr
a
phy
pr
im
itives,
where
ID
-
ba
sed
cou
l
d
be
subst
it
uted
f
or
ce
rtific
at
e
-
base
d
in
publi
c
key
set
ti
ng.
This
is
of
a
s
pe
ci
al
interest
pa
rtic
ularly
w
he
n
the
re
is
a
re
quirem
ent
for
ef
fici
ent
key
m
anag
e
m
e
nt
w
hile
m
od
er
at
e
secur
it
y
is
need
e
d.
The
e
nt
ire
dev
el
op
e
d
li
te
ratur
e
put
f
orward
on
ID
-
base
d
thres
ho
l
d
group
sig
natu
re
c
onta
ins
a
ppr
oac
hes
that
rely
on
an
in
div
id
ua
l
hard
pr
ob
le
m
su
ch
as
fac
toring,
Evaluation Warning : The document was created with Spire.PDF for Python.
IS
S
N
:
2088
-
8708
In
t J
Elec
&
C
om
p
En
g,
V
ol.
10
, No
.
4
,
A
ugus
t
2020 :
3695
-
3701
3696
discrete
l
og
a
rithm
or
el
li
ptic
curve
discrete
log
a
rithm
pr
ob
lem
[1
-
20]
.
He
reafter
,
if
a
s
ol
ution
of
a
ny
of
these
pro
blem
s
is
a
c
hieve
d,
the
n
the
secu
rity
of
the
ass
ociat
ed
I
D
-
base
d
th
reshold
gro
up
sign
at
ure
would
be
com
pr
om
ise
d.
Ther
e
f
or
e,
we
pr
ese
nt
in
t
his
work
a
secu
re
ID
-
bas
ed
t
hr
es
ho
l
d
gr
oup
sig
natu
re
inc
orp
orat
e
d
on
discrete
lo
gar
it
hm
s
and
r
esi
duos
it
y
pro
blem
.
Ou
r
te
chn
i
qu
e
s
en
hance
m
ent
arises
from
the
diff
ic
ulty
in
fin
ding
sim
ult
aneous
s
olu
ti
on
of
bot
h
pro
blem
s.
W
e
de
m
on
strat
e
that
our
a
ppro
ac
h
pe
rsists
to
be
secu
re,
des
pite
at
ta
ining
so
l
ution
of
one
of
t
he
pr
ob
le
m
s.
The
r
e
m
ai
nd
er
of
th
e
pa
per
is
struc
ture
d
as
fo
ll
ows:
the
IBTHS
is
intr
oduce
d
i
n
Sect
i
on
2.
Sect
io
n
3
pr
ese
nts
sec
ur
it
y
analy
sis
of
our
te
ch
nique.
Per
form
ance
stud
y
and
res
ultant e
ff
ic
ie
ncy a
re c
arr
ie
d o
ut in
Se
ct
ion
4.
Finall
y, we
conclu
de
i
n
Sect
io
n 5.
2.
THE
PROPO
SED IBTH
S
Her
e
,
we
will
intr
oduce
our
i
den
ti
ty
base
d
t
hr
es
hold
si
gn
at
ur
e
te
c
hniq
ue
that
reli
es
on
a
pair
of
ha
r
d
nu
m
ber
the
ore
ti
cal
pr
oble
m
s;
nam
el
y,
residu
os
it
y
an
d
dis
crete
log
a
rith
m
.
As
was
sta
te
d,
the
secu
rity
if
this
te
chn
iq
ue
bu
il
ds
on
the
pr
e
m
ise
that
i
t
is
burd
e
nsom
e
t
o
sim
ultaneousl
y
achieve
sol
uti
on
s
of
this
pair
of
pro
blem
s.
The
fr
am
ewo
r
k
of
our
te
c
hniq
ue
pr
es
um
es,
ou
t
of
si
gn
e
rs
are
able
to
jointl
y
sig
n
t
he
m
essag
e
on the acc
ount
of the
gro
up, w
her
eas
an i
ndiv
idu
al
ver
ifie
r
is
ab
le
to
corr
ob
or
at
e t
he gr
oup si
gnat
ur
e
[
21
].
2.1
.
S
ys
te
m se
tu
p
The
tr
us
te
d
de
al
er
(T
D)
of
the
syst
em
,
fo
ll
ow
in
g
t
he
f
ra
m
ewo
r
k
of
[
21
]
,
sel
ect
s
a
la
rg
e
pri
m
e
-
a
10
24
-
bits,
=
1
1
is
a
facto
r
of
−
1
,
w
her
e
1
an
d
1
are
two
s
afe
pr
im
es
,
an
el
em
ent
gen
erat
or
of
order
,
adh
e
rin
g t
o
≡
1
(
mod
)
, a
nd whe
r
e
ℎ
(
)
is t
he
one
-
w
ay
h
as
h
f
unct
io
n for the
m
essa
ge
.
2.2.
Gener
ati
ng
keys
W
it
hin
this
sta
ge,
TD
ca
rr
ie
s
-
out
the
conse
cutive
operati
ons
[
21
]
to
pro
du
ce
the
sec
re
t
and
public
keys
of
t
he
te
c
hn
i
qu
e:
-
Sele
ct
s in
a
ra
ndom
f
ashi
on
:
∈
∗
s
uch that
gcd
(
2
,
)
=
1
-
Ca
lc
ulate
≡
2
(
)
-
Af
te
r
that,
co
ns
tr
uct
a
(
,
)
t
hre
sho
ld
f
uncti
on
(
)
=
∘
+
1
+
⋯
+
−
1
−
1
(
mod
)
,
w
he
re
are r
a
ndom
integers bet
ween
1
an
d
−
1
, and
=
0
,
1
,
2
,
…
,
−
1
-
Set t
he gr
oup s
ecret key
(
0
)
, t
he
n
c
om
pu
te
the
associat
ed g
roup pu
blic key
=
(
0
)
(
mo
d
)
-
Each
of
the
gr
oup
m
e
m
ber
s
picks
an
integ
er
∈
(
)
in
a
rand
om
m
ann
er
as
his
pri
vate
ke
y
and
cal
culat
es h
is
public
key:
=
(
)
.
-
E
ach
par
ti
ci
pa
nt r
e
gisters a
n
i
den
ti
ty
an
d t
he
n
se
nds
to T
D
.
-
Af
te
r
T
D
ob
ta
i
ns
t
he
c
om
plete
identit
ie
s,
s/
he
cal
c
ulate
s:
(
)
an
d
=
(
)
(m
od
p
)
and
f
orwa
rds
(
)
to the
gr
oup’s
m
e
m
ber
s ind
i
vi
du
al
ly
. Pub
li
c
an
d reta
in a c
opy o
f
(
)
,
.
In
the
eve
nt
t
ha
t
an
e
xtra
m
e
m
ber
wis
hes
t
o
par
ti
ci
pate
within
the
gro
up
f
ollo
wing
ne
go
ti
at
io
n
with
t
he
TD
,
s
/he
posts
her
/
hi
s
ide
ntit
y
to
the
TD
.
Af
te
r
wh
ic
h,
T
D
cal
culat
es
a
nd
tra
ns
m
it
s
(
)
t
o
her
/
him
.
Then
,
TD
cal
c
ulate
s
an
d
pu
blics:
=
(
)
(
m
od
p
).
T
he
public
a
nd
secre
t
keys
f
or
an
i
nd
i
vidua
l
represe
nted
as
(
,
)
(
(
)
,
)
,
r
especti
vel
y.
Wh
il
e
for
gr
oup,
the
public
a
nd
secret
key
s
are
(
,
)
a
nd
(
,
(
0
)
)
, respecti
vely
.
2.3.
(
t,
n
)
Thr
esho
ld
sign
atu
re gener
ati
on
pha
se
Con
si
der
a
s
cenari
o
w
he
re
the
m
e
m
be
rs
that
c
oope
rate
in
produ
ci
ng
t
he
sig
na
ture
[21]
are
1
,
2
,
…
,
.
Ahead
to
their
c
ollabor
at
ive
sig
natu
re
of
t
he
m
essag
e,
a
sel
ect
ed
m
e
m
ber
is
a
ppoin
te
d
as
a
cl
erk
to
perf
or
m
par
ti
al
sign
at
ur
e
ve
rific
at
ion
.
The
seq
uen
ti
al
ste
ps
of
m
essage
sig
ning
a
re
il
lustrate
a
s
fo
ll
ows:
a.
Each si
gn
e
r
sel
ect
s
∈
∗
an
d
c
om
pu
te
s
≡
(
)
(1)
b.
The
{
}
broad
ca
ste
d
t
o
m
e
m
ber
s
by
m
eans
of
a
channel
t
hat
is
secur
e
.
When
entire
are
acq
uire
d,
t
he
y
are
util
iz
ed
col
le
ct
ively
in
the
co
m
pu
ta
ti
on of the
v
al
ue
as
=
∏
=
1
(
)
(2)
c.
Ca
lc
ulate
≡
(
ℎ
(
)
(
)
∏
−
−
=
1
,
≠
−
)
mod
(3)
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Elec
&
C
om
p
En
g
IS
S
N: 20
88
-
8708
Id
e
ntit
y
-
base
d t
hr
esh
old g
rou
p
si
gnatu
re
sc
he
me
based
on multi
ple
hard
num
ber
…
(
Ne
da
l
Tah
at
)
3697
T
he
n
se
nd
al
ong
with
(
,
)
as
the
par
ti
al
sig
na
ture
f
or
t
he
ha
sh
-
f
un
ct
io
n
m
essage
ℎ
(
)
to
th
e
cl
er
k.
Lat
er,
the
cl
er
k
pe
rfor
m
s
va
li
dation
of
the
par
ti
al
sign
at
ur
e
th
r
ough
dem
on
strat
in
g
that
the
subse
qu
e
nt
equ
al
it
y i
s fulfi
ll
ed:
=
ℎ
(
)
∏
(
–
−
)
=
1
,
≠
(
m
od
)
(4)
d.
Fo
ll
owin
g
dem
on
st
rati
ng the
validit
y of al
l p
arti
al
si
gn
at
ur
e
s b
y t
he
cl
er
k,
s/he
ob
ta
ins
s
ol
ution
f
or
:
2
=
−
2
∑
=
1
(
)
for
(5)
and the
sig
natu
re
of
m
essage
is
{
,
}
.
2.4.
(
t,
n
)
Thr
esho
ld
sign
atu
re
verific
at
i
on ph
as
e
The
sig
natu
re
can
be
ve
rified
by
a
strang
er p
en
d
in
g
that
s/he
can
get
a
hold
of
to
the
pu
blic
key
[2
1]
.
Fo
ll
owin
g
to
their
recepti
on
of the
gro
up si
gn
at
ur
e
{
,
}
s/
he
e
xam
ines the equati
on
:
2
=
ℎ
(
)
(
mod
)
(6)
If
t
his c
onditi
on is
fu
l
fill
ed,
a
ccordin
gly t
he gr
oup si
gnat
ur
e is
valid
.
Th
e
or
em
1.
S
uc
ceedin
g
the
ut
il
iz
ed
pr
ot
oc
ol
,
thu
s
t
he
ver
i
ficat
ion
within
the
sig
natu
re
ver
ific
at
io
n
pa
rt
is
accom
plished
.
2
≡
(
2
)
−
2
∑
=
1
(
mod
)
≡
∑
=
1
≡
∑
(
ℎ
(
)
(
)
∏
−
−
=
1
,
≠
−
)
=
1
∑
(
ℎ
(
)
(
)
∏
−
−
=
1
,
≠
)
=
1
−
∑
(
)
=
1
≡
(
∑
(
)
∏
−
−
=
1
,
≠
=
1
)
ℎ
(
)
(
∑
(
)
=
1
)
−
≡
(
(
0
)
)
ℎ
(
)
−
(
mo
d
)
≡
ℎ
(
)
(
m
od
)
(
7
)
3.
SECURIT
Y A
NA
L
YS
I
S
We d
em
on
st
rate her
e t
hat the
pr
ese
nted sche
m
e fo
r
i
den
ti
ty
–
ba
sed
t
hr
es
ho
ld sig
nature i
s u
nf
orgeab
l
e
fou
nd
on
the
c
om
plexity
of
f
ind
in
g
s
ol
utions
sim
ultaneou
s
to
the
pair
of
hard
nu
m
ber
t
heoreti
cal
pro
bl
e
m
s;
resid
uosit
y and discrete
log
a
rithm
. F
or
th w
hi
le
, w
e sh
al
l su
bs
ta
ntiat
e that our tech
nique i
s h
eu
risti
cal
ly
secur
e
against e
xam
ple crypt
ographi
c att
acks [2
1].
Att
ack
1
:
Sup
po
s
e
that
t
he
ad
ver
sa
ry
(Adv)
at
te
m
pts
to
acq
uire
the
secret
keys
,
(
0
)
ta
ken
f
r
om
the
eq
uations
≡
2
(
)
and
=
(
0
)
(
mo
d
)
.
It
is
evi
den
tl
y
infeasi
bl
e
in
view
of
t
he
hi
ndra
nce
of
fig
ur
in
g
ou
t
re
siduosit
y
and
di
screte
log
a
rith
m
pr
oble
m
s.
Also
A
dv
ca
nnot
der
ive
t
he
secr
et
key
(
)
fro
m
the equat
io
n
=
(
)
(
m
od
p
)
b
y
virt
ue of
the c
om
plica
ti
on
of s
olvi
ng
DL
P.
Att
ack 2
:
Assu
m
e that t
he discret
e lo
gar
it
hm
problem
can
be
f
ig
ur
e
d
-
out.
-
By
m
eans
of
t
he
e
qu
at
io
n
2
(
)
,
Adv
ca
n
fi
nd
2
(
)
.
Noneth
el
ess
,
s/
he
is
sti
ll
incap
able
of
recoveri
ng
be
cause
of the a
dversit
y o
f
s
olv
i
ng r
esi
duos
it
y
pro
blem
.
-
Adv
m
ay
li
kew
ise
at
te
m
pt
t
o
fig
ur
e
-
out
t
he
e
ntire
sec
r
et
keys
of
the
sig
ner
util
iz
ing
the
relat
io
ns
hi
p
=
(
)
(m
od
p
)
.
Co
nsi
der
in
g
that
discrete
lo
gar
i
thm
can
be
figure
d
-
ou
t,
at
that
po
i
nt
s/h
e
ca
n
disco
ver al
l secret
k
ey
s
(
)
, and
ther
ea
fter c
onstruct
the
e
ntir
e p
arti
al
sig
natur
e
of the
gro
up. Alt
houg
h,
Evaluation Warning : The document was created with Spire.PDF for Python.
IS
S
N
:
2088
-
8708
In
t J
Elec
&
C
om
p
En
g,
V
ol.
10
, No
.
4
,
A
ugus
t
2020 :
3695
-
3701
3698
s/he
is
un
a
ble
to
identify
the
gro
up
si
gn
at
ure
throu
gh
t
he
relat
ion
s
hip
2
=
−
2
∑
=
1
(
)
due
to the fact
that
s/he is
no
t a
wa
re
of
t
he prim
e
f
act
ori
zat
ion o
f
.
Att
ack
3
:
Assu
m
e
that
the
res
iduosit
y
pro
ble
m
is
so
lvable.
In
this
sit
uatio
n,
s/he
has
knowle
dge
of
the
pr
im
e
factor
iz
at
io
n
,
1
a
nd
1
.
Conseq
ue
ntly
,
s/he
will
at
tem
pt
to
fi
gure
-
out
the
form
ula
≡
2
(
)
.
Nev
e
rtheless
, s/
he
rem
ai
ns
incap
able
of
fig
uri
ng
-
out
rely
ing
on th
is c
ondi
ti
on
due to
t
he fact
that s/he is
n
ot
awar
e
of
2
(
)
f
or
t
he reaso
n
t
hat
a d
is
crete l
ogar
it
h
m
p
r
ob
le
m
can no
t
be u
nr
a
veled.
Att
ack
4
:
A
dv
can
in
a
ddit
ion
at
te
m
pt
to
gathe
r
pairs
of
m
essage
–
sign
at
ur
e
(
,
)
an
d
wh
e
re
=
1
,
2
,
…
,
al
so
trie
s
t
o
se
ek
-
out
the
i
nd
i
vidual
secret
ke
y
(
)
.
Me
an
w
hile,
A
dv
po
s
sess
es
eq
uatio
ns
i
n
this fas
hion:
1
≡
(
ℎ
(
1
)
(
)
∏
−
−
=
1
,
≠
−
1
1
)
2
≡
(
ℎ
(
2
)
(
)
∏
−
−
=
1
,
≠
−
2
2
)
≡
(
ℎ
(
)
(
)
∏
−
−
=
1
,
≠
−
)
(
8
)
The
nu
m
ber
o
f
unknow
ns
is
(
+
2
)
in
the p
re
vious
form
ulas
of
(
8
)
,
i.e.
,
(
)
,
an
d
.
Hen
c
e
,
(
)
an
d
rem
ai
n
co
m
plex
to
ide
nt
ify
co
ns
ide
rin
g
t
hat
Adv
c
an
re
pro
du
ce
a
li
m
it
ed
nu
m
ber
of
s
olu
ti
ons
to this set
of li
near eq
uatio
ns
al
tho
ug
h
s/
he
i
s not ca
pab
le
of ide
ntifyi
ng which is
the
rig
ht
o
ne
.
Att
ack
5:
A
dv
cou
l
d
seek
t
o
pose
as
sig
ner
by
ch
oo
si
ng
i
n
a
rando
m
fashi
on
inte
ger
s
and
a
nnou
ncin
g
g
≡
(
)
.
Be
cause
th
e
entire
signe
rs
decide
d
on
th
e
group
s
ign
at
ure,
in
the
a
bs
e
nce
of
knowle
dge
of
the
res
pecti
ve
secret
key
(
)
,
A
dv
is
i
ncap
a
bl
e
of
ge
ner
at
in
g
a
c
orrect
pa
rtia
l
sign
at
ure
(
,
)
to
fu
lfil
l t
he
verific
at
ion f
orm
ula.
Att
ack
6
:
Adv
m
a
y
con
te
nd
to
e
vo
l
ve
a
gro
up
sig
natu
re
(
,
)
of
his
own
us
in
g
t
he
ver
if
yi
ng
e
qu
at
i
on
2
=
ℎ
(
)
(
)
f
or
a
s
pecific
m
essage
thr
ough
fixing
one
inte
ger
,
whil
e
seekin
g
to
identif
y
the
ot
her.
I
n
this
sce
nar
i
o,
A
dv
picks
an
d
see
ks
t
he
value
of
.
A
dv
beg
i
ns
by
cal
culat
i
ng
≡
ℎ
(
)
−
(
)
an
d
fi
nd
i
ng
≡
2
(
)
for
.
Un
s
uccess
f
ully
,
s/he
is
una
bl
e
to
fi
gure
-
out
util
iz
ing
this
e
qu
at
io
n
becaus
e
of
th
e
ad
ver
s
it
y
of
fig
ur
i
ng
-
ou
t,
sim
ultaneou
sly
,
t
he
resi
duos
it
y
an
d
di
screte
log
a
rithm
pr
oblem
s.
In
ad
diti
on,
A
dv
co
uld
at
tem
pt
to
set
and
fig
ur
e
-
out
.
In
this
sit
uatio
n,
s/he
c
om
pu
te
s
≡
ℎ
(
)
−
2
(
)
an
d
see
ks
ou
t
a
so
luti
on
t
o
≡
(
)
.
T
h
is
sce
nari
o
is
the
w
or
s
t
du
e
to
the
fact
that
al
thou
gh
bot
h
prob
le
m
s
of
resi
duos
it
y
an
d
dis
crete
lo
gar
it
hm
can
be
s
olv
e
d,
the
val
ue
rem
ai
ns
diff
ic
ult
to
determ
ine
asi
de
fr
om
a
tria
l
and
error
pr
ocedure
,
thu
s
cha
ract
erized
by
co
nsum
ption
of
tim
e
and
effor
t
[15].
4.
PERFO
R
MANC
E E
V
ALU
TION
To
in
vestigat
e
the
pe
rform
ance
of
i
de
ntit
y
-
based
thres
hold
si
gnat
ur
e
,
c
om
puta
ti
on
an
d
com
m
un
ic
at
ion
ov
e
rh
e
ads
w
il
l
be
us
e
d
to
est
i
m
at
e
i
t.
He
re,
we
will
ex
a
m
ine
pr
im
arily
the
pe
rfor
m
ance
of
our
sug
gested
te
chn
iq
ue.
T
o
facil
it
at
e
this
t
reatm
ent,
we
e
m
plo
y
the
fo
ll
ow
i
ng
no
ta
ti
on
s
in
our
analy
s
is
of
the
com
pu
ta
ti
on
an
d
c
omm
un
ic
at
ion
com
plexity
[2
2
-
25
]
.
T
he
num
ber
of
secret
an
d
pu
bl
ic
keys
are
de
no
te
d
by
S
K
a
nd
P
K,
res
pecti
vel
y.
Mo
du
la
r
e
xpone
ntiat
ion
ti
m
e
is
represe
nt
ed
by;
,
w
hile
sta
nds
f
or
the
tim
e
fo
r
m
od
ular
m
ultip
li
cat
ion
;
an
d
the
tim
e
fo
r
a
m
od
ular
in
ve
rse
c
om
pu
ta
tio
n
is
re
pr
ese
nt
ed
by
;
de
note
s
c
om
plexity
of
tim
e
fo
r
exe
c
uting
c
om
pu
ta
ti
on
of
t
he
m
odular
s
qu
a
re;
presents
the
c
om
plexity
f
or
exec
utin
g
cal
culat
ion
o
f
t
he
m
od
ula
r
s
quare
r
oo
t
.
ℎ
dete
rm
ines
the
one
m
ap
-
to
-
po
i
nt h
as
h
functi
on
ti
m
e,
wh
il
e
|
|
s
pecifie
s
the
le
ngth
of
bits
of
.
We
m
us
t
no
te
that
ot
her
com
pu
t
at
ion
al
op
e
rati
on
s
tim
es
are
igno
r
ed,
since
t
hey
are
m
uch
sm
aller
tha
n
,
,
a
nd
ℎ
.
We
su
m
m
ari
ze
the
c
om
pu
ta
ti
on
and
c
omm
un
ic
at
ion
cost
of
our
pro
posed
schem
e
in
Table
1.
As
show
n
in
Ta
ble
1
,
the
com
pu
ta
ti
on
com
plexity
[22]
f
or
si
gn
at
ure
an
d
ve
rificat
ion
a
re
4
+
(
4
2
+
+
1
)
+
(
2
−
+
1
)
+
+
+
ℎ
and
3
+
+
+
ℎ
in
our
schem
e,
respe
ct
ively
.
Als
o
the
t
otal
c
om
m
un
i
cat
ion
cost
a
r
e
(
2
+
1
)
|
|
+
(
3
+
1
)
|
|
.
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Elec
&
C
om
p
En
g
IS
S
N: 20
88
-
8708
Id
e
ntit
y
-
base
d t
hr
esh
old g
rou
p
si
gnatu
re
sc
he
me
based
on multi
ple
hard
num
ber
…
(
Ne
da
l
Tah
at
)
3699
T
able
1
.
Th
e
pe
rfor
m
ance of
our
sc
hem
e
Criteria
Evalu
atio
n
No
.
o
f
keys
SK
2
+
1
PK
2
t+1
Co
m
p
u
tatio
n
al co
m
p
l
ex
ity
Sig
n
4
+
(
4
2
+
+
1
)
+
(
2
−
+
1
)
+
+
+
ℎ
Verify
3
+
+
+
ℎ
Size of
para
m
e
ters
/co
m
m
u
n
icatio
n
c
o
st
(
2
+
1
)
|
|
+
(
3
+
1
)
|
|
5.
ILL
US
TR
AT
ION
To
il
lustrate
the
pr
op
os
e
d
schem
e,
we
con
s
ider
that
there
are
three
us
e
rs
.
The
truste
d
de
al
er
(TD)
of
t
he
syst
em
cho
ose
=
14447
,
1
=
31
,
2
=
233
,
t
he
n
=
1
2
=
7223
,
=
8
a
nd
ℎ
(
)
=
801
.
The follo
wing
ste
p
il
lustrate
our
sch
em
e.
5.1
.
Ke
ys
gene
rat
i
on
In this ste
p,
t
he
subse
qu
e
nt act
ion
s
are
ca
rr
ie
d
-
out by T
D
t
o ge
ner
at
e t
he
s
chem
e’s
secret
and
public k
ey
s:
a.
Sele
ct
s
in
a
ra
ndom
fas
hion
=
223
∈
∗
a
nd
obser
ving
that
gcd
(
2
,
)
=
1
.
Ca
lc
ulate
≡
8
63
9
1
m
od
14447
≡
8853
mod
14447
.
Sele
ct
a
poly
no
m
ia
l
(
)
=
311
+
733
+
123
2
(
mod
7223
)
.
Set
the
gro
up
sec
ret
key
(
0
)
=
311
an
d
cal
c
ulate
s
the
c
orres
pond
in
g
gro
up
public
key
=
8
311
mo
d
14447
≡
10022
mod
14447
.
b.
Each
of
the
three
m
e
m
ber
s
of
the
gr
oup,
in
a
ra
ndom
fash
i
on,
picks
an
intege
r
as
:
1
=
163
,
2
=
237
,
3
=
757
,
to
re
pr
e
s
ent
his
pri
va
te
key,
a
nd
then
dete
r
m
ines
his
public
key
1
=
8
163
m
od
14447
≡
5619
,
2
≡
8
237
m
od
14447
≡
2811
,
3
≡
8
757
m
od
14447
≡
1670
.
c.
Each
pa
rtic
ipa
nt
reg
ist
er
s
a
n
i
de
ntit
y
1
=
321
,
2
=
531
,
3
=
239
an
d
t
he
n
sen
ds
to
T
D.
TD
c
om
pu
te
s a
nd pu
blics
1
≡
(
1
)
m
od
≡
8
2146
mod
14447
≡
8300
2
≡
(
2
)
m
od
≡
8
3072
mo
d
14447
≡
8450
3
≡
(
3
)
m
od
≡
8
50
mo
d
14447
≡
1543
5.
2.
(
t,
n
)
Thr
esho
ld
sign
atu
re gener
ati
on
pha
se
Assum
e
that
t
he
m
e
m
ber
s
par
ti
ci
pating
in
the
signa
ture
ge
ner
at
io
n
are
1
,
2
,
…
,
.
Pr
ece
ding
to
j
oi
ntly
sign
i
ng
the
m
essage,
on
e
of
t
hese
m
e
m
ber
s
is
a
pp
oi
nted
as
a
cl
e
r
k
to
perform
ver
ific
at
ion
the
par
ti
al
sign
at
ur
e.
W
e
descr
i
be
the
el
e
m
ents o
f
m
es
sage s
i
gn
i
ng in
seque
nce as f
ol
lows
:
Each si
gn
e
r
sel
ect
s
1
=
11
7
,
2
=
147
,
3
=
371
an
d
c
om
pu
te
s
1
≡
8
16
3
(
11
7
)
mo
d
14447
≡
10094
2
≡
8
237
(
1
47
)
mo
d
14447
≡
2746
3
≡
8
757
(
3
71
)
mo
d
14447
≡
1247
The
{
}
are
br
oa
dcasted
th
r
ough
a
sec
ure
c
ha
nn
el
to
the
m
e
m
ber
s.
Sub
seq
uen
t
to
rec
eption
of
the
e
ntire
,
each
of
them
com
pu
te
s
the
value
as
=
∏
3
=
1
(
)
=
(
10094
×
2746
×
1247
)
mod
≡
9787
Ca
lc
ulate
s
1
=
5195
,
2
=
3174
,
3
=
1583
.
The
n
se
nd
al
ong
with
(
,
)
,
repr
esenti
ng
t
he
ha
sh
-
functi
on
m
e
ssage
ℎ
(
)
par
ti
al
si
gn
at
ur
e
,
t
o
t
he
cl
erk
.
Subseque
nt
to
validat
io
n
of
e
ntire
pa
r
ti
al
sign
at
ur
es
by
the
cl
erk
,
s/he
deter
m
ines
so
luti
on
of
2
=
−
2
∑
=
1
(
)
for
,
2
=
1172
(
2894
)
mod
7223
≡
4181
.
5.3.
(
,
)
Thre
sh
ol
d sig
n
ature
ve
ri
ficat
io
n pha
se
An
y ne
wc
om
e
r
is able t
o
per
f
or
m
v
erifica
ti
on
o
f
the sig
nat
ur
e gra
nted
tha
t s/
he
can access the p
ubli
c
key. Succee
ding to
h
is
r
ece
ption o
f
t
he gr
oup si
gnat
ur
e
,
{
,
}
he revie
ws:
2
=
ℎ
(
)
(
mod
)
2
≡
8
853
4181
9787
9
78
7
(
m
od
14447
)
≡
3768
ℎ
(
)
≡
10022
801
(
mod
14447
)
≡
3768
Evaluation Warning : The document was created with Spire.PDF for Python.
IS
S
N
:
2088
-
8708
In
t J
Elec
&
C
om
p
En
g,
V
ol.
10
, No
.
4
,
A
ugus
t
2020 :
3695
-
3701
3700
6.
CONCL
US
I
O
N
A
ne
w
te
ch
ni
qu
e
f
or
ID
-
ba
sed
th
res
ho
l
d
gro
up
si
gn
at
ure
was
pr
opose
d,
wh
ic
h
is
fou
nd
e
d
on
the
prob
le
m
s
of
resid
uosit
y
a
nd
discrete
loga
rithm
.
The
tech
ni
qu
e
reli
es
on
tw
o
dif
ficul
t
har
d
pro
blem
s
and
offer
s
an
im
pr
ov
e
d
le
vel
of
secur
it
y
relat
iv
e
to
a
n
in
div
i
dual
di
ff
ic
ult
prob
le
m
.
Also,
we
hav
e
in
ves
ti
gated
so
m
e
po
te
ntial
at
ta
cks
an
d
dem
on
strat
ed
th
e
secu
rity
of
t
he
sc
hem
e
against
su
c
h
at
ta
c
ks
.
I
n
a
dd
it
io
n,
the
schem
e
is
resi
st
ant
to
both
of
re
peat
an
d
con
s
piracy
at
ta
cks.
M
or
e
ov
e
r,
each
of
gr
oup
sign
at
ur
e
a
nd
gro
up
key size
s
do no
t rely
on th
e
nu
m
ber
o
f
m
e
m
ber
s
.
REFERE
NCE
S
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W
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Diffie,
M.
E
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lman
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dire
c
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ptogr
aph
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at
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and
b
li
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nat
ure
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m
es,
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nti
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"
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for
ide
n
ti
t
y
b
ase
d
ide
nt
ifica
t
i
on
and
sign
at
ur
e
sche
m
es,
"
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roce
edi
ngs
o
f
I
n
te
rnational
Co
nfe
renc
e
on
th
e
Theory
and
Appl
ic
a
ti
ons
o
f
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e
rla
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eh,
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M.
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in,
"
Ide
nti
t
y
-
B
ase
d
en
c
r
y
pt
ion
from
th
e
W
ei
l
p
ai
r
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"
In
i
t
Crypto
2001
,
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ta
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ara,
US
A,
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29,
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F.
Li
,
J.
Yu,
"
A
New
Thre
shold
Group
Signat
ur
e
Scheme
Based
on
Discre
t
e
Lo
gar
it
hm
Proble
m
,
"
Ei
ghth
ACI
S
Inte
rnational
Co
nfe
renc
e
N
et
work
and
Parallel/
D
istribut
ed
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uti
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B
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Yan
g
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Y.
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"
Ide
nt
ity
-
ba
sed
thr
eshold
signa
ture
an
d
m
edi
ated
prox
y
sign
at
ur
e
sch
e
m
es,
"
J
ournal
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f
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Univ
ersities of
Posts and Tele
communic
a
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07.
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Huawang,
Z.
Xiaohua,
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Y
uewe
i,
"
Provabl
y
se
cur
e
Id
ent
i
t
y
-
bas
ed
thr
eshold
signat
ure
on
a
cc
ess
struct
u
re,
"
Inte
rnational
Co
nfe
renc
e
on
Info
rm
ati
on
and
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mm
unic
ati
ons Tec
hnolog
ie
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4
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L.
D
eng
and
J.
Ze
ng
,
"
Two
ne
w
ide
nt
ity
-
b
ase
d
thre
shold
ring
s
igna
tur
e
sch
emes,
"
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[16]
N.
Ta
h
at,
"
Convert
ible
m
ul
ti
-
a
uthe
ntica
te
d
encr
y
pt
ion
sch
eme
with
ver
i
ficati
on
base
d
on
elliptic
cur
ve
d
iscret
e
loga
rit
hm
prob
lem
,
"
Int.
J. Comp
ute
r A
pp
li
ca
ti
on
s in
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logy,
vol
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no
.
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p
p.
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-
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,
201
6.
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S.
Chang,
D.
S.
W
ong,
Y
.
Mu
,
Z
.
Zha
ng
,
"
Cert
if
ic
a
te
l
ess
Th
reshold
Ring
Si
gnat
ure
,
"
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ie
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–
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J.
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Z.
Ch
en,
F.
L
i
,
"
On
the
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ur
ity
of
an
ide
nt
ity
b
ase
d
m
ult
i
-
prox
y
sign
at
ure
sche
m
e,
"
Computers
and
Elec
tri
cal E
ngine
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2011.
[19]
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Fag
en,
H.
Y
upu,
C
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Jie
,
"
Im
prove
m
ent
of
Ide
nt
ity
.
B
ase
d
Thre
shold
Prox
y
Sign
a
tur
e
Sch
eme
with
Kno
wn
Signers,
"
W
uhan
Unive
rs
ity
Journal
of
Natural
S
ci
en
ce
s,
vol
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no
.
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-
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Y.
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L.
Fa
gen,
X.
Chunxi
ang,
S.
Ying
,
"
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Eff
i
cient
Id
ent
ity
-
B
ase
d
An
on
y
m
ous
Signcr
y
pt
ion
Scheme,
"
Wuhan
Unive
rs
ity
Journal
of
Nat
ural
Scienc
es
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v
ol
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no
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ad
a
nd
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.
Ism
ai
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"
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y
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d
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,
"
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n
te
rnational
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f
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ai
l
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N.
M.F.
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h
at
,
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R.
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Ahm
ad,
"
A
New
Digit
al
Signat
ure
Sch
e
m
e
Based
on
Fa
ct
oring
and
Disc
ret
e
Loga
ri
thms
,
"
J
o
urnal
of
Ma
the
m
ati
cs
and
Statis
tics
,
vol
.
4
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no
.
4
,
pp.
222
-
225
,
20
08.
[23]
N.
Ta
h
at
,
R
.
S
haqboua
,
E.
Ab
dal
l
ah,
M.
Bso
ul
and
W
asfi
S
hat
an
awi,
"
A
n
ew
digi
t
al
signa
ture
sche
m
e
wi
t
h
m
essage
rec
ov
e
r
y
using
h
y
brid
proble
m
s,
"
Int
ernati
onal
Journal
of
El
e
ct
ri
c
al
and
Comput
er
Engi
n
ee
ring
,
"
vol
.
9
,
no
.
5
,
pp
.
3576
-
3583,
201
9.
[24]
N.
T
aha
t
and
M
.
S.
Hij
az
i
,
"
A
New
Digital
Signat
ure
Scheme
Based
on
Chaotic
Map
s
and
Q
uadr
atic
Residu
e
Problems
,
"
A
ppl
ie
d
Ma
the
mati
cs
&
Information
Sci
en
ce
s
,
vol
.
13
,
no
.
1
,
pp
.
115
-
1
20,
2019
.
[25]
N.
Ta
h
at,
A.
A
.
Ta
h
at
,
M.
Abu
-
Dalu,
R
.
B
.
Alb
ada
rne
h
,
A.
E
.
Abdall
ah
and
O.
M.
Al
-
Haz
ai
m
e
h,
"
A
new
RS
A
publi
c
ke
y
e
ncr
ypti
on
sche
m
e
wi
th
cha
ot
ic
m
aps
,
"
Inte
rnational
J
ournal
of
E
lectri
cal
and
Comput
er
Eng
ineering
(
IJE
CE)
,
vol. 10
,
no
.
2
,
pp
.
1430
-
1437,
2020
.
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Elec
&
C
om
p
En
g
IS
S
N: 20
88
-
8708
Id
e
ntit
y
-
base
d t
hr
esh
old g
rou
p
si
gnatu
re
sc
he
me
based
on multi
ple
hard
num
ber
…
(
Ne
da
l
Tah
at
)
3701
BIOGR
AP
HI
ES OF
A
UTH
ORS
Nedal
Tah
at
re
c
ei
v
ed
his
BS
c
i
n
Mathe
m
atics
f
rom
Yarm
ouk
U
nive
rsit
y
,
Jordan
in
1994,
and
MS
c
in
Pure
M
at
hemat
ic
s
a
t
Al
al
-
Ba
y
t
Unive
rsit
y
,
Jordan
,
in
1998.
He
rec
eived
a
PhD
in
Applie
d
Num
ber
The
or
y
(Cr
y
pto
gra
ph
y
)
from
Na
ti
onal
Univer
si
t
y
of
Mal
a
y
si
a
(U
KM
)
in
2010.
H
e
is
an
As
soci
at
e
Profess
or
in
the
Depa
r
tment
Mathe
m
at
i
cs,
H
ashe
m
it
e
Univer
sit
y
.
His
m
ai
n
rese
arc
h
in
te
r
est
s
are
cr
y
p
tol
og
y
and
num
ber
the
or
y
.
He
has
publi
shed
m
ore
tha
n
35
pape
rs,
aut
hore
d
/c
oau
th
ore
d,
and
m
ore
t
han
15
r
efe
r
ee
d
j
ourna
l and conference pa
p
ers
As
hraf
A.
Tah
a
t
is
an
As
socia
te
Profess
or
in
the
Depa
rtment
of
Com
m
unic
at
ions
Engi
nee
r
ing
at
Pr
incess
Suma
y
a
Univ
ersity
f
or
Technol
og
y
(
PS
UT)
and
the
Vice
-
Cha
irman
of
IEEE
Jordan
Sect
ion
.
Dr.
T
ah
at
e
arn
ed
his
B.
Sc.
and
M.Sc
.
d
egr
ee
s
in
El
e
ct
r
i
ca
l
Engi
ne
eri
ng
from
the
Ill
ino
is
Instit
ute
of
Te
ch
nolog
y
(I
ll
ino
isTe
ch)
,
Chicago,
US
A,
where
he
al
so
rec
e
iv
ed
a
Ph.D.
in
2002
,
with
a
foc
us
on
comm
unic
at
ion
s
and
signa
l
pro
ce
ss
ing.
Dr
.
T
a
hat
joi
ned
PS
UT
in
2005
and
serve
d
as
the
Hea
d
of
the
depa
r
t
m
ent
of
Comm
u
nic
a
ti
ons E
ng.
fr
om
2010
to
2012
.
He
was a
lso a
Visiti
ng
Profess
or
with
McGill
Univer
sit
y
,
M
ontre
a
l,
Ca
n
ada,
in
the
Depa
r
t
m
ent
of
ECE
,
conduc
t
ing
rese
arc
h
on
m
oder
n
comm
unic
at
ion
s
sy
st
ems
(2012
-
2013).
From
2002
to
2003,
h
e
was a
n
Adjun
ct
Profess
or
at
I
ll
in
oisTe
ch
,
Chi
ca
g
o,
US
A.
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